A violin bow has a screw that allows you to adjust the hairs' tension. Like a tuning peg so to speak.

When I just got mine, I didn't pay attention to it. I worked with the tuning pegs because I knew I would need to, but I didn't touch the violin screw because, well, I have no idea why should I.

Tutorials and such don't seem to talk much about it. All I understood from a quick read around is that it is kind of an early improvement to the bow. This suggests me that it really isn't very necessary in the first place.

I think I also heard that a "proper" tension level is when you can pass a finger through the gap, no more and no less.

2 Answers
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The hair can (and should) be loosened when not playing so that the bow is not constantly subjected to hair tension. It's the same reason some people recommend loosening guitar strings when storing a guitar for a longer period of time, except a guitar neck has the advantage of a stiff metal rod inside it, which the violin bow does not.

It allows hair tension to be adjusted to suit one's playing style, as well as certain techniques like spiccato (bouncing the bow off the strings). While guidelines like "the gap should be the width of a finger" (some also say a pencil) are well-known, they're not hard rules, as different players might prefer a different amount of tension.

It also allows hair tension to be adjusted in response to changes in air temperature and humidity, which affect both the hair and the wood of the bow, and thus the playability of the instrument.

It can certainly be adjusted for different playing styles and conditions as mentioned already, and this is definitely something to bear in mind.

However, the primary reason is really to preserve the condition of the bow when not in use, and is why you should always loosen your bow off when you're not using it. Constantly subjecting the wooden rod of the bow to tension will likely cause it to prematurely warp and twist, which is initially just a bit annoying but can eventually lead to uneven tension along the length of the bow and an uneven balance, making it incredibly hard to play and essentially rendering the bow a write-off. Not a great thing when you consider some bows easily stretch into the thousands of pounds range.

Yes, you could make the argument for loosening strings on instruments as well, but generally I'd argue against that, the instruments are designed to take the tension this way for extended periods. The wood on the bow is not, and generally being much thinner and deliberately more flexible than say a violin neck, is much more subject to abuse.